Integration and Accumulation of Change
Core ideas
The definite integral is a limit of Riemann sums and measures net accumulation. The Fundamental Theorem of Calculus links integration and differentiation: $\int_a^b f(x),dx=F(b)-F(a)$ when $F'=f$, and $\frac{d}{dx}\int_a^{g(x)}f(t),dt=f(g(x))g'(x)$. Accumulation functions model net change of a rate.
Techniques include u-substitution (reverse chain rule), integration by parts $\int u,dv=uv-\int v,du$, and partial fractions for rational functions. Properties of definite integrals - linearity, additivity over intervals, and even/odd symmetry - simplify many computations. Average value is $\frac{1}{b-a}\int_a^b f(x),dx$.
Traps
- Dropping the differential in substitution or forgetting to change bounds when substituting in a definite integral.
- Sign errors in the FTOC when the variable upper limit is inside a composition; the chain factor is required.
- Confusing net accumulation with total distance; total distance sums absolute values.
- Integration by parts requires choosing $u$ wisely (LIATE heuristic) and may need repetition.
Worked example
Compute $\frac{d}{dx}\int_0^{x^2}\sin t,dt$. By FTOC with chain rule, this equals $\sin(x^2)\cdot2x$. The inner derivative is essential: forgetting it yields the wrong answer.